The problem which group one were unsuccessful in solving was one which considered
area. The task was to work out the possible dimensions of a rectangular swimming pool of
which the area was 36 metres squared. The students were told that the pool was not 9 x 4 metres and they had to compile possible alternative solutions: 1 x 36, 2 x 18, 3 x 12, 6 x 6. The teacher suggested that this would be a difficult task for the students due to the need to be able to differentiate between area and perimeter. Both of these concepts had been covered relatively recently.
Over the three sessions which were recorded, group one generally displayed positive interactions. However, they were prone to large portions of off-task talk. According to the teacher, at least two members of the group should have been able to solve the problem if working on their own and so it had been expected that they would have achieved the solution as a group. The group was not successful and on viewing the video data, it became clear that there was one particular point which might have proved critical in the solution process.
The portion of video which was shown to the group was from a point when one member of the group, Scott, appeared to have worked out the solution with the help of the teacher. The teacher then requested that Scott share this information with the rest of the group, and she left. However, the rest of the group seemed less than willing to listen to Scott, and he refused to interact with them until they displayed behaviours which he felt were
acceptable. In this section the group members are displaying contradictions in the rules they apply to social interaction.
Engestrom (1993) suggested that rules could be both explicit and implicit rules which individuals use to guide their actions and interactions. Group one in the study displayed contradictions in the rules that governed the willingness of group members to engage in collaborative metacognition. The following text, taken from the problem solving session highlights this.
244 Teacher Ok, fantastic. What do you know about this rectangle (referring to the sheet).
245 Scott it’s 36 metres squared.
247 Duncan Because 9 times 4 is 36.
248 Teacher Ok, if that’s wrong, what have you got to do?
249 Lara 4 times 9?
250 Scott No!
251 Duncan thirty, thirty, thirty-six divided by four?
252 Lara Which is nine!
253 Duncan Thirty six divided by nine?
254 Scott Oh No!!
255 Teacher No, anything else?
256 Scott No you have to, with 36 , these bits are smaller
(making a rectangular shape with his hands) so you need…. 36 metres altogether these bits (to opposite sides) are, we have to figure out two things, the two ones (sides) that are shorter and the two ones (sides) that are longer and make up 36.
257 Teacher Fantastic!
258 Duncan What?
259 Teacher Say what you just said, that’s exactly what you’ve got to do.
In the above excerpt there are two points where there was collaborative metacognition between the teacher and a student. At line 247, Duncan was able to explain his reasoning behind his understanding of the problem, he also extended Scott’s contribution of 36 metres squared. At line 256, Scott is able to explain to the teacher what he thinks he has to do in order to solve the problem, whilst correcting Duncan’s suggestion. At the end of the interaction, the teacher encouraged Scott to share his thinking with the rest of the group and then left. In doing so, she presumably hoped that Scott would display metacognition in order to draw the group in to the discussion. This was a particularly critical point in the solution process as Scott had seemed to misunderstand the problem and was trying to work out the perimeter of the rectangle instead of the area. Had he explained his thought
processes to the group members, there may have been the potential for them to engage in a discussion with Scott in order to correct his thinking. This type of interaction would have resulted in student-student collaborative metacognition.
willingness of students to engage in collaborative metacognition, the result of which was that Scott’s misunderstanding was not highlighted until much later in the session:
261 Scott The two smaller bits, the two smaller bits needs to be the same length. Guys! (Duncan and Riana are laughing and joking together rather than listening to Scott).
262 Duncan Yeah?
263 Scott Mrs McKenzie just told me to explain it again. 264 Duncan Keep explaining. Keep, keep explaining 265 Riana (laughs and says something about her fingers) 266 Scot I won’t until Riana stops laughing
267 Duncan Finlay… I mean… (laughs) Scott, Scott keep explaining
268 Scott not until Riana stops laughing.
269 Duncan (hides Riana behind his pencil case) Riana’s not there any more. Scott just explain, just explain.
270 Scott No
271 Duncan Explain
272 Scott I’m not
273 Duncan I’ll tell Mrs McKenzie.
274 Scott I’m just gonna blame Riana then. 275 Riana How can you blame me?
276 Scott Cause I’m not doing it cause you are laughing. 277 Duncan C’mon just explain it.
278 Scott Well I am now, right.
279 Riana (inaudible)
280 Duncan Scott!! Scott just do it!
281 Lara Riana stop laughing, you’re making me laugh! 282 Riana Lara’s making me laugh.
283 Duncan Just explain Scott, come on, hurry up I’m gonna get so bored!
The interactions above suggest a certain amount of tension between Scott and the rest of the students. Scott was adhering to certain social rules, whereby he would not explain to the rest of the group unless they stopped laughing. This interaction went on for some time and, to a large extent, contributed to the lack of success. Whilst this text clearly shows Scott’s reluctance to explain the problem when the group was not giving him their
attention, evidence from the recall interview highlights a deeper historical problem. Scott reported a dislike for working with others and a belief that he would learn more if he worked on his own. These implicit rules regarding the type of people Scott is willing to interact with on a level which requires collaborative metacognition seemed to be affecting his ability to interact during this problem:
Researcher Can you tell me what was going on at this point in the film – what were you thinking?
Scott They wouldn’t stop giggling and it was getting me annoyed
Researcher Scott, what did you think when Mrs McKenzie asked you to explain the problem to the rest of the group – were you able to explain the problem? Scott not really, they were all giggling.
Researcher how did you all feel working in the group together – did you enjoy that? Scott I prefer to work on my own because you don’t have to explain things
Researcher What’s wrong with explaining things – did you know in your head what you wanted to say?
Scott I could work it out much quicker on my own Researcher did you enjoy explaining it to Mrs McKenzie? Scott yes
Researcher and you looked really happy when she said you got it right so why couldn’t you share that with other people – could you still remember the solution? Scott well it depends who it is – if it was John and Mark (his two friends) I
wouldn’t mind cause they would listen but they just all kept giggling at me. Researcher so why would John and Mark listen?
Scott cause they’re on the same level as me.
Scott reported here that he enjoyed explaining his thinking to the teacher. He also enjoyed explaining it to those he perceived to be on the same level as he is. Scott’s rules for interacting in a manner which would result in collaborative metacognition were based on his perceptions of the intellectual abilities of others.
This preference to work individually rather than in groups was echoed by Duncan and Lara. Duncan’s rules for deciding who he was willing to interact with were based on intellectual ability. However, Lara, preferred to work with friends. Only one member of the group, Riana, said she preferred to work with others. Duncan, Lara and Scott also viewed the group as a place where individual learning was less likely to occur:
Researcher Scott feels it’s quite difficult to explain things to people because it slows him down
Duncan it is
Research and do you find that? Duncan I do
Researcher so would you rather work in a group or on your own? Duncan on my own – I like working individually
Scott me too
Riana that’s what Scott says but I don’t like working individually because I can never work it out
Researcher so you quite like working in a group Riana because you feel you can’t work it out on your own?
Scott but you can’t do it in a group if you keep on giggling
Researcher but the group can get the answer right and you’ve maybe not learned anything
Scott exactly – so it’s better if you do it individually because you do learn something
Researcher but what if you did it in a group and you all learned something? Lara well in that (video) situation what would have helped?
Duncan if Scott had actually explained the problem
The text presented above shows that the willingness of students to make their thinking known to others was influenced by implicit rules which each individual created regarding who they were willing to interact with. By implicit, I mean that they were not made
explicit during the problem solving session. They also had clear opinions on the likelihood of learning in a group situation. Three out of four of the group members believed that learning was less likely to occur in a group situation.
students used to guide their interactions was highlighted. Scott began to explain his thoughts on the problem to the other group members. As I mentioned previously, Scott was trying to work out the perimeter of the rectangle rather than the area. To do so, he was assigning measurements to the sides of the rectangle which might produce 36 (which referred to the area). Riana attempted to interact and tried to make helpful suggestions. However, Scott was unwilling to listen to her suggestions and kept cutting her off:
348 Scott Right, these two sides are smaller than these two sides (makes shapes with his hands) so they, let me get this right.
349 Lara Why is the long one 3 and the short one 15?
350 Scott Because it’s not supposed to be, it’s not exactly. 15 and 15 and 3 equals 33 and 3, 36. So that’s how long the rope should be.
351 Riana Why don’t you get..(Scott cuts her off)
352 Scott These two sides are shorter than these two sides (make shapes with his hands) and it’s 36
353 Riana Why don’t you get..(Scott cuts her off)
354 Scott and these two sides should be 3cm, 3 metres, and these two sides should be 15 metres.
355 Riana Why don’t you get a big rope that’s 36metres area 356 Duncan and put it round
357 Lara Aah (David hit her foot with the chair) 358 Riana And put it roun…(Scott cuts her off)
359 Scott We’ve factored [sic] the problem out! Hazzah!!
As previously mentioned, the difficulty in this task was that the students had to
differentiate between perimeter and area. Scott was wrong in his choice and began to try and work out the perimeter. I believe that contradictions in rules for interactions displayed by the students contributed to the lack of collaborative metacognition. Had the students interacted in such a way, they may have realised that Scott was on the wrong track with his suggestions. However, Scott’s rules which prevented him from interacting with the group when they were laughing, also prevented him from interacting when other group members made suggestions. Scott had suggested that he preferred to work on his own because explaining took too long. Furthermore, if he had to work with others he preferred to work
with those on the same level as he is. Scott specifically referred to other class members whom he believed to be on the same level and did not include those in his group. He also noted that these class members would not engage in the behaviours displayed by those members of his group, such as constant laughing. Therefore, had Scott been in a group which contained members which fitted into his rules for engagement, then he may have engaged in the kind of interactions which result in collaborative metacognition.
Towards the end of the session, the teacher joined the group and realised that Scott had been working on the perimeter rather than the area. Despite the teacher correcting the students and suggesting that the group work together to find the area rather than the perimeter, it was only Scott who made any attempt to do so. Furthermore, he made no attempt to involve the rest of the group, who were interacting on a social level. The overall result was that the group was unsuccessful in completing the task.